StandardSimplex

The standard n-simplex, denoted Δ^n, is a canonical n-dimensional polytope used in geometry, topology, and numerical analysis. It can be defined as the set of points t in R^{n+1} with nonnegative coordinates that sum to 1: Δ^n = { t ∈ R^{n+1} : t_i ≥ 0 for i = 0,..., n and ∑ t_i = 1 }. It is the convex hull of the standard basis vectors e_0, ..., e_n, so its n+1 vertices are these unit vectors.

Equivalently, Δ^n can be viewed in R^n as Δ^n = { x ∈ R^n : x_i ≥ 0 for all i and ∑ x_i ≤ 1 }, via the affine embedding t_i = x_i for i ≤ n and t_{n+1} = 1 − ∑ x_i. These two representations are affinely isomorphic and hence describe the same combinatorial object.

The simplex is a compact, convex, n-dimensional polytope. Its faces correspond to setting some coordinates to zero, yielding all lower-dimensional simplices. In the standard representation, Δ^n has n+1 vertices (the e_i) and 2^{n+1} faces in total.

In the usual R^n representation, the n-dimensional volume of Δ^n is 1/n!. The object also appears as a canonical model in probability theory, where the coordinates represent a probability distribution: the probability simplex.

The standard simplex is invariant under permutations of coordinates, giving it a high degree of symmetry. It serves as a fundamental building block in simplicial complexes, triangulations, and various algorithms in computational geometry and optimization.